En mathématiques, une variété kählérienne ou variété de Kähler est une variété différentielle équipée d'une structure unitaire satisfaisant une condition d'intégrabilité. C'est en particulier une variété riemannienne, une variété symplectique et une variété complexe, ces trois structures étant mutuellement compatibles. Les variétés kählériennes sont un objet d'étude naturel en géométrie différentielle complexe. Elles doivent leur nom au mathématicien Erich Kähler. Définition Plusieurs définitions équivalentes existent. Cela est dû aux relations entre structures complexes, symplectiques et riemanniennes. Une manière de le comprendre est de constater que le groupe unitaire U(n) (qui joue le rôle de groupe de structure d'une variété kählérienne) est l'intersection d'un couple quelconque des trois groupes GL_n(\Complex), Sp(n) et O(2n). Donnons deux définitions équivalentes.

Une variété kählérienne est une variété hermi

Fundamental Group And Pluridifferentials On Compact Kahler Manifolds

Yohan Cédric Loup Brunebarbe

A compact Kahler manifold X is shown to be simply connected if its 'symmetric cotangent algebra' is trivial. Conjecturally, such a manifold should even be rationally connected. The relative version is also shown: a proper surjective connected holomorphic map f : X -> S between connected manifolds induces an isomorphism of fundamental groups if its smooth fibres are as above, and if X is Kahler.

Independent Univ Moscow-Ium2016

Vacuum Stability in Supersymmetric Effective Theories

Leonardo Brizi

The main topics discussed in this thesis are supersymmetric low-energy effective theories and metastability conditions in generic non-renormalizable models with global and local supersymmetry. In the first part we discuss the conditions under which the low-energy expansion in space-time derivatives preserves supersymmetry implying that heavy multiplets can be more efficiently integrated out directly at the superfield level. These conditions translate into the requirements that also fermions and auxiliary fields should be small compared to the heavy mass scale. They apply not only to the matter sector, but also to the gravitational one if present, and imply in that case that the gravitino mass should be small. We finally give a simple prescription to integrate out heavy chiral and vector superfields consisting respectively in imposing stationarity of the superpotential and of the Kähler potential; the procedure holds in the same form both for global and local supersymmetry. In the second part we study general criteria for the existence of metastable vacua which break global supersymmetry in models with local gauge symmetries. In particular we present a strategy to define an absolute upper bound on the mass of the lightest scalar field which depends on the geometrical properties of the Kähler target manifold. This bound can be saturated by properly tuning the superpotential and its positivity therefore represents a necessary and sufficient condition for the existence of metastable vacua. It is derived by looking at the subspace of all those directions in field space for which an arbitrary supersymmetric mass term is not allowed and scalar masses are controlled by supersymmetry-breaking splitting effects. This subspace includes not only the direction of supersymmetry breaking, but also the directions of gauge symmetry breaking and the lightest scalar is in general a linear combination of fields spanning all these directions. Our purpose is to show that the largest value for the lightest mass is in general achieved when the lightest scalar is a combination of the Goldstone and the Goldstino partners. We conclude by computing the effects induced by the integration of heavy multiplets on the light masses. In particular we focus on the sGoldstino partners and we show that heavy chiral multiplets induce a negative level-repulsion effect that tends to compromise vacuum stability, whereas heavy vector multiplets in general induce a positive-definite contribution. Our results find application in the context of string-inspired supergravity models, where metastability conditions can be used to discriminate among different compactification scenarios and supersymmetric effective theories can be used to face the problem of moduli stabilization.

EPFL2011

Tackling the Supersymmetric Flavour Problem in String Models

Christopher John Andrey

In this work we address one of the phenomenological issues of beyond the Standard Model scenarios which embed Supersymmetry, namely the Supersymmetric Flavour Problem, in the context of String Theory. Indeed, the addition of new interactions to the Standard Model generically spoils its flavour structure which is one of its major achievements since it for example leads to a very elegant understanding of the absence of flavour changing neutral currents in the leptonic sector and of the stability of the proton, thanks to accidental symmetries. We focus on a subset of the phenomenologically dangerous operators, namely the soft scalar masses. One way out of the Supersymmetric Flavour Problem is to geographically separate the observable and hidden sectors along a fifth dimension, gravity being the only interaction propagating in the bulk. In such scenarios, the soft scalar masses are vanishing at the classical level since there is no direct contact term between the observable and hidden multiplets and tend to be universal at the loop-level. However such setups hardly ever come about in String Theory, which is one of the most promising candidates of quantum gravity. In order to make contact with the five-dimensional picture, we focus on the prototypical case of the E8 × E8 Heterotic M-Theory which, in a certain regime, effectively looks five-dimensional and embeds matter fields on two end-of-the-world branes. In these scenarios, not only gravity but also vector multiplets propagate in the five-dimensional bulk, effectively spoiling the sequestered picture. However, since the contact terms responsible for the appearance of soft scalar masses arise due to the exchange of heavy vectors, they do enjoy a current-current structure which can be exploited to inhibit the emergence of soft scalar masses by postulating a global symmetry in the hidden sector. In order to assess the possibility of realising such a mechanism, we first study the full dependence of the Kähler potential on both the moduli and the matter fields in the case of orbifold and Calabi-Yau compactifications. We then determine whether an effective sequestering may be achieved thanks to a global symmetry and argue that whereas for orbifold models our strategy can naturally be put at work, it can only be implemented in a subset of Calabi-Yau models.

EPFL2011

Vacuum Stability in Supersymmetric Effective Theories

Leonardo Brizi

EPFL2011

Tackling the Supersymmetric Flavour Problem in String Models

Christopher John Andrey

EPFL2011

Variété kählérienne

Une variété kählérienne est une variété hermi

Fundamental Group And Pluridifferentials On Compact Kahler Manifolds

Vacuum Stability in Supersymmetric Effective Theories

Tackling the Supersymmetric Flavour Problem in String Models

Vacuum Stability in Supersymmetric Effective Theories

Fundamental Group And Pluridifferentials On Compact Kahler Manifolds

Tackling the Supersymmetric Flavour Problem in String Models